19 ideas
| 14782 | Philosophy is an experimental science, resting on common experience [Peirce] |
| Full Idea: Philosophy, although it uses no microscopes or other apparatus of special observation, is really an experimental science, resting on that experience which is common to us all. | |
| From: Charles Sanders Peirce (The Nature of Mathematics [1898], I) | |
| A reaction: The 'experimental' either implies that thought-experiments are central to the subject, or that philosophers are discussing the findings of scientists, but at a high level of theory and abstraction. Peirce probably means the latter. I can't disagree. |
| 14787 | Self-contradiction doesn't reveal impossibility; it is inductive impossibility which reveals self-contradiction [Peirce] |
| Full Idea: It is an anacoluthon to say that a proposition is impossible because it is self-contradictory. It rather is thought so to appear self-contradictory because the ideal induction has shown it to be impossible. | |
| From: Charles Sanders Peirce (The Nature of Mathematics [1898], III) |
| 11064 | Classes can be reduced to propositional functions [Russell, by Hanna] |
| Full Idea: Russell held that classes can be reduced to propositional functions. | |
| From: report of Bertrand Russell (Mathematical logic and theory of types [1908]) by Robert Hanna - Rationality and Logic 2.4 | |
| A reaction: The exact nature of a propositional function is disputed amongst Russell scholars (though it is roughly an open sentence of the form 'x is red'). |
| 14783 | Logic, unlike mathematics, is not hypothetical; it asserts categorical ends from hypothetical means [Peirce] |
| Full Idea: Mathematics is purely hypothetical: it produces nothing but conditional propositions. Logic, on the contrary, is categorical in its assertions. True, it is a normative science, and not a mere discovery of what really is. It discovers ends from means. | |
| From: Charles Sanders Peirce (The Nature of Mathematics [1898], II) |
| 6407 | The class of classes which lack self-membership leads to a contradiction [Russell, by Grayling] |
| Full Idea: The class of teaspoons isn't a teaspoon, so isn't a member of itself; but the class of non-teaspoons is a member of itself. The class of all classes which are not members of themselves is a member of itself if it isn't a member of itself! Paradox. | |
| From: report of Bertrand Russell (Mathematical logic and theory of types [1908]) by A.C. Grayling - Russell Ch.2 | |
| A reaction: A very compressed version of Russell's famous paradox, often known as the 'barber' paradox. Russell developed his Theory of Types in an attempt to counter the paradox. Frege's response was to despair of his own theory. |
| 14788 | Mathematics is close to logic, but is even more abstract [Peirce] |
| Full Idea: The whole of the theory of numbers belongs to logic; or rather, it would do so, were it not, as pure mathematics, pre-logical, that is, even more abstract than logic. | |
| From: Charles Sanders Peirce (The Nature of Mathematics [1898], IV) | |
| A reaction: Peirce seems to flirt with logicism, but rejects in favour of some subtler relationship. I just don't believe that numbers are purely logical entities. |
| 10418 | Type theory seems an extreme reaction, since self-exemplification is often innocuous [Swoyer on Russell] |
| Full Idea: Russell's reaction to his paradox (by creating his theory of types) seems extreme, because many cases of self-exemplification are innocuous. The property of being a property is itself a property. | |
| From: comment on Bertrand Russell (Mathematical logic and theory of types [1908]) by Chris Swoyer - Properties 7.5 | |
| A reaction: Perhaps it is not enough that 'many cases' are innocuous. We are starting from philosophy of mathematics, where precision is essentially. General views about properties come later. |
| 10047 | Russell's improvements blocked mathematics as well as paradoxes, and needed further axioms [Russell, by Musgrave] |
| Full Idea: Unfortunately, Russell's new logic, as well as preventing the deduction of paradoxes, also prevented the deduction of mathematics, so he supplemented it with additional axioms, of Infinity, of Choice, and of Reducibility. | |
| From: report of Bertrand Russell (Mathematical logic and theory of types [1908]) by Alan Musgrave - Logicism Revisited §2 | |
| A reaction: The first axiom seems to be an empirical hypothesis, and the second has turned out to be independent of logic and set theory. |
| 23478 | Type theory means that features shared by different levels cannot be expressed [Morris,M on Russell] |
| Full Idea: Russell's theory of types avoided the paradoxes, but it had the result that features common to different levels of the hierarchy become uncapturable (since any attempt to capture them would involve a predicate which disobeyed the hierarchy restrictions). | |
| From: comment on Bertrand Russell (Mathematical logic and theory of types [1908]) by Michael Morris - Guidebook to Wittgenstein's Tractatus 2H |
| 21718 | Ramified types can be defended as a system of intensional logic, with a 'no class' view of sets [Russell, by Linsky,B] |
| Full Idea: A defence of the ramified theory of types comes in seeing it as a system of intensional logic which includes the 'no class' account of sets, and indeed the whole development of mathematics, as just a part. | |
| From: report of Bertrand Russell (Mathematical logic and theory of types [1908]) by Bernard Linsky - Russell's Metaphysical Logic 6.1 | |
| A reaction: So Linsky's basic project is to save logicism, by resting on intensional logic (rather than extensional logic and set theory). I'm not aware that Linsky has acquired followers for this. Maybe Crispin Wright has commented? |
| 18126 | A set does not exist unless at least one of its specifications is predicative [Russell, by Bostock] |
| Full Idea: The idea is that the same set may well have different canonical specifications, i.e. there may be different ways of stating its membership conditions, and so long as one of these is predicative all is well. If none are, the supposed set does not exist. | |
| From: report of Bertrand Russell (Mathematical logic and theory of types [1908]) by David Bostock - Philosophy of Mathematics 8.1 |
| 18128 | Russell is a conceptualist here, saying some abstracta only exist because definitions create them [Russell, by Bostock] |
| Full Idea: It is a conceptualist approach that Russell is relying on. ...The view is that some abstract objects ...exist only because they are definable. It is the definition that would (if permitted) somehow bring them into existence. | |
| From: report of Bertrand Russell (Mathematical logic and theory of types [1908]) by David Bostock - Philosophy of Mathematics 8.1 | |
| A reaction: I'm suddenly thinking that predicativism is rather interesting. Being of an anti-platonist persuasion about abstract 'objects', I take some story about how we generate them to be needed. Psychological abstraction seems right, but a bit vague. |
| 18124 | Vicious Circle says if it is expressed using the whole collection, it can't be in the collection [Russell, by Bostock] |
| Full Idea: The Vicious Circle Principle says, roughly, that whatever involves, or presupposes, or is only definable in terms of, all of a collection cannot itself be one of the collection. | |
| From: report of Bertrand Russell (Mathematical logic and theory of types [1908], p.63,75) by David Bostock - Philosophy of Mathematics 8.1 | |
| A reaction: This is Bostock's paraphrase of Russell, because Russell never quite puts it clearly. The response is the requirement to be 'predicative'. Bostock emphasises that it mainly concerns definitions. The Principle 'always leads to hierarchies'. |
| 14786 | Some logical possibility concerns single propositions, but there is also compatibility between propositions [Peirce] |
| Full Idea: Many say everything is logically possible which involves no contradiction. In this sense two contradictory propositions may be severally possible. In the substantive sense, the contradictory of a possible proposition is impossible (if we were omniscient). | |
| From: Charles Sanders Peirce (The Nature of Mathematics [1898], III) |
| 14789 | Experience is indeed our only source of knowledge, provided we include inner experience [Peirce] |
| Full Idea: If Mill says that experience is the only source of any kind of knowledge, I grant it at once, provided only that by experience he means personal history, life. But if he wants me to admit that inner experience is nothing, he asks what cannot be granted. | |
| From: Charles Sanders Peirce (The Nature of Mathematics [1898]) | |
| A reaction: Notice from Idea 14785 that Peirce has ideas in mind, and not just inner experiences like hunger. Empiricism certainly begins to look more plausible if we expand the notion of experience. It must include what we learned from prior experience. |
| 14785 | The world is one of experience, but experiences are always located among our ideas [Peirce] |
| Full Idea: The real world is the world of sensible experience, and it is part of the process of sensible experience to locate its facts in the world of ideas. | |
| From: Charles Sanders Peirce (The Nature of Mathematics [1898], III) | |
| A reaction: This is the neatest demolition of the sharp dividing line between empiricism and rationalism that I have ever encountered. |
| 14784 | Ethics is the science of aims [Peirce] |
| Full Idea: Ethics is the science of aims. | |
| From: Charles Sanders Peirce (The Nature of Mathematics [1898], II) | |
| A reaction: Intriguing slogan. He is discussing the aims of logic. I think what he means is that ethics is the science of value. 'Science' may be optimistic, but I would sort of agree with his basic idea. |
| 541 | Virtue comes more from habit than character [Critias] |
| Full Idea: More men are good through habit than through character. | |
| From: Critias (fragments/reports [c.440 BCE], B09), quoted by John Stobaeus - Anthology 3.29.41 |
| 542 | Fear of the gods was invented to discourage secret sin [Critias] |
| Full Idea: When the laws forbade men to commit open crimes of violence, and they began to do them in secret, a wise and clever man invented fear of the gods for mortals, to frighten the wicked, even if they sin in secret. | |
| From: Critias (fragments/reports [c.440 BCE], B25), quoted by Sextus Empiricus - Against the Professors (six books) 9.54 |